| L(s) = 1 | − 3.74e12·2-s + 2.81e20·3-s − 2.46e25·4-s − 8.27e29·5-s − 1.05e33·6-s + 1.89e35·7-s + 2.37e38·8-s + 4.35e40·9-s + 3.09e42·10-s + 1.68e44·11-s − 6.96e45·12-s + 2.68e47·13-s − 7.07e47·14-s − 2.33e50·15-s + 6.86e49·16-s − 3.29e52·17-s − 1.63e53·18-s + 2.01e54·19-s + 2.04e55·20-s + 5.33e55·21-s − 6.32e56·22-s − 4.49e57·23-s + 6.68e58·24-s + 4.26e59·25-s − 1.00e60·26-s + 2.16e60·27-s − 4.66e60·28-s + ⋯ |
| L(s) = 1 | − 0.601·2-s + 1.48·3-s − 0.638·4-s − 1.62·5-s − 0.894·6-s + 0.229·7-s + 0.985·8-s + 1.21·9-s + 0.979·10-s + 0.930·11-s − 0.949·12-s + 1.22·13-s − 0.137·14-s − 2.42·15-s + 0.0458·16-s − 1.67·17-s − 0.729·18-s + 0.907·19-s + 1.03·20-s + 0.340·21-s − 0.559·22-s − 0.602·23-s + 1.46·24-s + 1.65·25-s − 0.733·26-s + 0.317·27-s − 0.146·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(86-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+85/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(43)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{87}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3.74e12T + 3.86e25T^{2} \) |
| 3 | \( 1 - 2.81e20T + 3.59e40T^{2} \) |
| 5 | \( 1 + 8.27e29T + 2.58e59T^{2} \) |
| 7 | \( 1 - 1.89e35T + 6.81e71T^{2} \) |
| 11 | \( 1 - 1.68e44T + 3.29e88T^{2} \) |
| 13 | \( 1 - 2.68e47T + 4.84e94T^{2} \) |
| 17 | \( 1 + 3.29e52T + 3.87e104T^{2} \) |
| 19 | \( 1 - 2.01e54T + 4.94e108T^{2} \) |
| 23 | \( 1 + 4.49e57T + 5.58e115T^{2} \) |
| 29 | \( 1 + 1.32e62T + 2.01e124T^{2} \) |
| 31 | \( 1 + 3.19e63T + 5.83e126T^{2} \) |
| 37 | \( 1 + 1.15e66T + 1.98e133T^{2} \) |
| 41 | \( 1 - 1.04e68T + 1.22e137T^{2} \) |
| 43 | \( 1 - 5.39e68T + 6.99e138T^{2} \) |
| 47 | \( 1 + 1.33e71T + 1.34e142T^{2} \) |
| 53 | \( 1 - 2.43e73T + 3.65e146T^{2} \) |
| 59 | \( 1 + 4.35e74T + 3.32e150T^{2} \) |
| 61 | \( 1 - 7.94e73T + 5.66e151T^{2} \) |
| 67 | \( 1 + 2.81e77T + 1.64e155T^{2} \) |
| 71 | \( 1 + 7.15e78T + 2.27e157T^{2} \) |
| 73 | \( 1 - 4.09e78T + 2.41e158T^{2} \) |
| 79 | \( 1 - 8.75e79T + 1.98e161T^{2} \) |
| 83 | \( 1 + 6.07e81T + 1.32e163T^{2} \) |
| 89 | \( 1 - 1.38e82T + 4.99e165T^{2} \) |
| 97 | \( 1 - 3.11e84T + 7.50e168T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.67035118544939173445060776565, −13.33200027677734090973691218157, −11.26988592215776210880511277661, −9.129975766229432350768006731215, −8.434651534857655608698189288078, −7.38353066478613388457772202799, −4.20167196612041258851446831494, −3.55661286206166526449441616560, −1.52923498664387286713313153080, 0,
1.52923498664387286713313153080, 3.55661286206166526449441616560, 4.20167196612041258851446831494, 7.38353066478613388457772202799, 8.434651534857655608698189288078, 9.129975766229432350768006731215, 11.26988592215776210880511277661, 13.33200027677734090973691218157, 14.67035118544939173445060776565
